Best Known (231−49, 231, s)-Nets in Base 3
(231−49, 231, 688)-Net over F3 — Constructive and digital
Digital (182, 231, 688)-net over F3, using
- t-expansion [i] based on digital (181, 231, 688)-net over F3, using
- 1 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 58, 172)-net over F81, using
- 1 times m-reduction [i] based on digital (181, 232, 688)-net over F3, using
(231−49, 231, 1942)-Net over F3 — Digital
Digital (182, 231, 1942)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3231, 1942, F3, 49) (dual of [1942, 1711, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 2208, F3, 49) (dual of [2208, 1977, 50]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3230, 2207, F3, 49) (dual of [2207, 1977, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(3211, 2188, F3, 45) (dual of [2188, 1977, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(35, 19, F3, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3230, 2207, F3, 49) (dual of [2207, 1977, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3231, 2208, F3, 49) (dual of [2208, 1977, 50]-code), using
(231−49, 231, 183100)-Net in Base 3 — Upper bound on s
There is no (182, 231, 183101)-net in base 3, because
- 1 times m-reduction [i] would yield (182, 230, 183101)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 54 692654 187282 671095 667790 133787 851819 828164 662925 248987 098625 519581 562025 975797 825024 446101 204095 477053 556001 > 3230 [i]